Bimoment

Bimoment
Bimoment
	Mω
Units
SI	kgm ²
GHS	g cm ²

The moment ( deforming moment ) ^[1] - a physical quantity , bending- torque , is formed when the profile is loaded at an angle or with an uneven load on the profile.

This term is used in the analysis of beams ( continuum mechanics ), refers to torsion and deformation, and is denoted by Mω ^[2] . The beam moment shows the distribution in the cross section of the (longitudinal) strain stress for cases of strain twisting and strain distortion, respectively ^[3] . As a rule, a moment can be represented by a pair of equal and opposite bending moments ^[4] .

Content

1 Attitude to tension
2 Relation to movements
3 Calculation of the beam moment
4 See also
5 notes
6 Literature
7 References

Attitude to Tension

The obtained bimoment on the site can be calculated as the integral of the product of unitary strain and stress, perpendicular to the cross section:

B_{\omega }(x)=\int _{A}\omega (y,z)\sigma _{x}(x,y,z)\ dydz

{\ displaystyle B _ {\ omega} (x) = \ int _ {A} \ omega (y, z) \ sigma _ {x} (x, y, z) \ dydz}

Relation Relationships

The moment can be considered as a generalized averaged strain force φ (deformation function). To prove this, it suffices to consider the expression of the strain energy for a subjected to bending torsion:

e_{def}=e_{flex}+e_{tor}+e_{fl-tr}\;

{\ displaystyle e_ {def} = e_ {flex} + e_ {tor} + e_ {fl-tr} \;}

Where each of these conditions is expressed in terms of generalized displacements of the barycentric axis and derivatives of these displacements. Direct check:

B_{\omega }={\frac {\partial e_{def}}{\partial (d\varphi /dx)}}=0+{\frac {\partial e_{tor}}{\partial (d\varphi /dx)}}+0=EI_{\omega }{\frac {d\varphi }{dx}}

{\ displaystyle B _ {\ omega} = {\ frac {\ partial e_ {def}} {\ partial (d \ varphi / dx)}} = 0 + {\ frac {\ partial e_ {tor}} {\ partial ( d \ varphi / dx)}} + 0 = EI _ {\ omega} {\ frac {d \ varphi} {dx}}}

Where only the energy condition is used, the torque is determined as follows:

e_{tor}={\frac {1}{2}}\left[GJ\left({\frac {d\theta _{x}}{dx}}\right)^{2}+{\frac {\kappa }{1-\kappa }}GJ\left({\frac {d\theta _{x}}{dx}}-\varphi \right)^{2}+EI_{\omega }\left({\frac {d\varphi }{dx}}\right)^{2}\right]

{\ displaystyle e_ {tor} = {\ frac {1} {2}} \ left [GJ \ left ({\ frac {d \ theta _ {x}} {dx}} \ right) ^ {2} + { \ frac {\ kappa} {1- \ kappa}} GJ \ left ({\ frac {d \ theta _ {x}} {dx}} - \ varphi \ right) ^ {2} + EI _ {\ omega} \ left ({\ frac {d \ varphi} {dx}} \ right) ^ {2} \ right]}

Calculation of Beamment

The moment can be calculated from the stresses per unit length from a system of differential equations:

{\begin{cases}{\cfrac {d\varphi }{dx}}={\cfrac {B_{\omega }}{EI_{\omega }}}\\{\cfrac {dB_{\omega }}{dx}}=\kappa _{0}GJ\varphi -\phi (x;Q_{y},Q_{z},M_{x})\end{cases}}

{\ displaystyle {\ begin {cases} {\ cfrac {d \ varphi} {dx}} = {\ cfrac {B _ {\ omega}} {EI _ {\ omega}}} \\ {\ cfrac {dB _ {\ omega }} {dx}} = \ kappa _ {0} GJ \ varphi - \ phi (x; Q_ {y}, Q_ {z}, M_ {x}) \ end {cases}}}

Where:

J,I_{\omega }\;

{\ displaystyle J, I _ {\ omega} \;}

- torsion modulus and deformation modulus, respectively;

\kappa _{0}:=1-J/(I_{y}+I_{z})\;

{\ displaystyle \ kappa _ {0}: = 1-J / (I_ {y} + I_ {z}) \;}

- calculated by the torsion modulus and the polar moment of inertia or the sum of the main moments of inertia.

Having received the second of these equations and substituting the first relation into it, we obtain a second-order equation for the bimoment:

{\frac {d^{2}B_{\omega }}{dx^{2}}}-\kappa _{0}{\frac {GJ}{EI_{\omega }}}B_{\omega }=-{\frac {d\phi }{dx}}

{\ displaystyle {\ frac {d ^ {2} B _ {\ omega}} {dx ^ {2}}} - \ kappa _ {0} {\ frac {GJ} {EI _ {\ omega}}} B _ {\ omega} = - {\ frac {d \ phi} {dx}}}

Notes

↑ Theory of Beam-Columns, Volume 2: Space Behavior and Design . - P. 274.
↑ Wai-Fah Chen, Toshio Atsuta. Theory of Beam-Columns, Volume 2: Space Behavior and Design . - McGraw-Hill Inc., 2008.
↑ Methods of analysis and design of concrete boxbeams with side cantilevers: Technical report . - Cement and Concrete Association. - P. 1974.
↑ Proceedings - Institution of Civil Engineers: Research and theory, Volume 63 . - Institution of Civil Engineers, 1977. - P. 848.

Literature

Bychkov D.V. Structural mechanics of rod thin-walled structures / D.V. Bychkov . - VI-6513. - M .: Gosstroyizdat , 1952.- 475 p. - 5,500 copies, copies.
SP 16.13330.2011 (Updated version of SNiP II-23-81 *) "Steel structures".

Links

Calculation of runs taking into account bimoment
General case of loading a thin-walled rod
A. R. Tusnin, M. Prokich, “Strength of I- Beams in Constrained Torsion, Given the Development of Plastic Deformations” . Scientific and technical magazine on construction and architecture " Vestnik MGSU ". 2014. No 1. Page 75-82. ISSN 2304-6600 (Online), ISSN 1997-0935.
Structural Stability: Elastic, Inelastic, Destructive, and Destructive Theories

[1] Theory of Beam-Columns, Volume 2: Space Behavior and Design . - P. 274.

[2] Wai-Fah Chen, Toshio Atsuta. Theory of Beam-Columns, Volume 2: Space Behavior and Design . - McGraw-Hill Inc., 2008.

[3] Methods of analysis and design of concrete boxbeams with side cantilevers: Technical report . - Cement and Concrete Association. - P. 1974.

[4] Proceedings - Institution of Civil Engineers: Research and theory, Volume 63 . - Institution of Civil Engineers, 1977. - P. 848.